Building Monotonicity-Preserving Fuzzy Inference Models with Optimization-Based Similarity Reasoning and a Monotonicity Index

In this paper, a novel approach to building a Fuzzy Inference System (FIS) that preserves the monotonicity property is proposed. A new fuzzy re-labeling technique to re-label the consequents of fuzzy rules in the database (before the Similarity Reasoning process) and a monotonicity index for use in FIS modeling are introduced. The proposed approach is able to overcome several restrictions in our previous work that uses mathematical conditions in building monotonicity-preserving FIS models. Here, we show that the proposed approach is applicable to different FIS models, which include the zero-order Sugeno FIS and Mamdani models. Besides, the proposed approach can be extended to undertake problems related to the local monotonicity property of FIS models. A number of examples to demonstrate the usefulness of the proposed approach are presented. The results indicate the usefulness of the proposed approach in constructing monotonicity-preserving FIS models

Efficient Data Driven Multi Source Fusion

Data/information fusion is an integral component of many existing and emerging applications; e.g., remote sensing, smart cars, Internet of Things (IoT), and Big Data, to name a few. While fusion aims to achieve better results than what any one individual input can provide, often the challenge is to determine the underlying mathematics for aggregation suitable for an application. In this dissertation, I focus on the following three aspects of aggregation: (i) efficient data-driven learning and optimization, (ii) extensions and new aggregation methods, and (iii) feature and decision level fusion for machine learning with applications to signal and image processing. The Choquet integral (ChI), a powerful nonlinear aggregation operator, is a parametric way (with respect to the fuzzy measure (FM)) to generate a wealth of aggregation operators. The FM has 2N variables and N(2N − 1) constraints for N inputs. As a result, learning the ChI parameters from data quickly becomes impractical for most applications. Herein, I propose a scalable learning procedure (which is linear with respect to training sample size) for the ChI that identifies and optimizes only data-supported variables. As such, the computational complexity of the learning algorithm is proportional to the complexity of the solver used. This method also includes an imputation framework to obtain scalar values for data-unsupported (aka missing) variables and a compression algorithm (lossy or losselss) of the learned variables. I also propose a genetic algorithm (GA) to optimize the ChI for non-convex, multi-modal, and/or analytical objective functions. This algorithm introduces two operators that automatically preserve the constraints; therefore there is no need to explicitly enforce the constraints as is required by traditional GA algorithms. In addition, this algorithm provides an efficient representation of the search space with the minimal set of vertices. Furthermore, I study different strategies for extending the fuzzy integral for missing data and I propose a GOAL programming framework to aggregate inputs from heterogeneous sources for the ChI learning. Last, my work in remote sensing involves visual clustering based band group selection and Lp-norm multiple kernel learning based feature level fusion in hyperspectral image processing to enhance pixel level classification

Fuzzy Sets, Fuzzy Logic and Their Applications 2020

The present book contains the 24 total articles accepted and published in the Special Issue “Fuzzy Sets, Fuzzy Logic and Their Applications, 2020” of the MDPI Mathematics journal, which covers a wide range of topics connected to the theory and applications of fuzzy sets and systems of fuzzy logic and their extensions/generalizations. These topics include, among others, elements from fuzzy graphs; fuzzy numbers; fuzzy equations; fuzzy linear spaces; intuitionistic fuzzy sets; soft sets; type-2 fuzzy sets, bipolar fuzzy sets, plithogenic sets, fuzzy decision making, fuzzy governance, fuzzy models in mathematics of finance, a philosophical treatise on the connection of the scientific reasoning with fuzzy logic, etc. It is hoped that the book will be interesting and useful for those working in the area of fuzzy sets, fuzzy systems and fuzzy logic, as well as for those with the proper mathematical background and willing to become familiar with recent advances in fuzzy mathematics, which has become prevalent in almost all sectors of the human life and activity

Personalizable Knowledge Integration

Large repositories of data are used daily as knowledge bases (KBs) feeding computer systems that support decision making processes, such as in medical or financial applications. Unfortunately, the larger a KB is, the harder it is to ensure its consistency and completeness. The problem of handling KBs of this kind has been studied in the AI and databases communities, but most approaches focus on computing answers locally to the KB, assuming there is some single, epistemically correct solution. It is important to recognize that for some applications, as part of the decision making process, users consider far more knowledge than that which is contained in the knowledge base, and that sometimes inconsistent data may help in directing reasoning; for instance, inconsistency in taxpayer records can serve as evidence of a possible fraud. Thus, the handling of this type of data needs to be context-sensitive, creating a synergy with the user in order to build useful, flexible data management systems. Inconsistent and incomplete information is ubiquitous and presents a substantial problem when trying to reason about the data: how can we derive an adequate model of the world, from the point of view of a given user, from a KB that may be inconsistent or incomplete? In this thesis we argue that in many cases users need to bring their application-specific knowledge to bear in order to inform the data management process. Therefore, we provide different approaches to handle, in a personalized fashion, some of the most common issues that arise in knowledge management. Specifically, we focus on (1) inconsistency management in relational databases, general knowledge bases, and a special kind of knowledge base designed for news reports; (2) management of incomplete information in the form of different types of null values; and (3) answering queries in the presence of uncertain schema matchings. We allow users to define policies to manage both inconsistent and incomplete information in their application in a way that takes both the user's knowledge of his problem, and his attitude to error/risk, into account. Using the frameworks and tools proposed here, users can specify when and how they want to manage/solve the issues that arise due to inconsistency and incompleteness in their data, in the way that best suits their needs

ISIPTA'07: Proceedings of the Fifth International Symposium on Imprecise Probability: Theories and Applications

B

Feature and Decision Level Fusion Using Multiple Kernel Learning and Fuzzy Integrals

The work collected in this dissertation addresses the problem of data fusion. In other words, this is the problem of making decisions (also known as the problem of classification in the machine learning and statistics communities) when data from multiple sources are available, or when decisions/confidence levels from a panel of decision-makers are accessible. This problem has become increasingly important in recent years, especially with the ever-increasing popularity of autonomous systems outfitted with suites of sensors and the dawn of the ``age of big data.\u27\u27 While data fusion is a very broad topic, the work in this dissertation considers two very specific techniques: feature-level fusion and decision-level fusion. In general, the fusion methods proposed throughout this dissertation rely on kernel methods and fuzzy integrals. Both are very powerful tools, however, they also come with challenges, some of which are summarized below. I address these challenges in this dissertation. Kernel methods for classification is a well-studied area in which data are implicitly mapped from a lower-dimensional space to a higher-dimensional space to improve classification accuracy. However, for most kernel methods, one must still choose a kernel to use for the problem. Since there is, in general, no way of knowing which kernel is the best, multiple kernel learning (MKL) is a technique used to learn the aggregation of a set of valid kernels into a single (ideally) superior kernel. The aggregation can be done using weighted sums of the pre-computed kernels, but determining the summation weights is not a trivial task. Furthermore, MKL does not work well with large datasets because of limited storage space and prediction speed. These challenges are tackled by the introduction of many new algorithms in the following chapters. I also address MKL\u27s storage and speed drawbacks, allowing MKL-based techniques to be applied to big data efficiently. Some algorithms in this work are based on the Choquet fuzzy integral, a powerful nonlinear aggregation operator parameterized by the fuzzy measure (FM). These decision-level fusion algorithms learn a fuzzy measure by minimizing a sum of squared error (SSE) criterion based on a set of training data. The flexibility of the Choquet integral comes with a cost, however---given a set of N decision makers, the size of the FM the algorithm must learn is 2N. This means that the training data must be diverse enough to include 2N independent observations, though this is rarely encountered in practice. I address this in the following chapters via many different regularization functions, a popular technique in machine learning and statistics used to prevent overfitting and increase model generalization. Finally, it is worth noting that the aggregation behavior of the Choquet integral is not intuitive. I tackle this by proposing a quantitative visualization strategy allowing the FM and Choquet integral behavior to be shown simultaneously

Learning from samples using coherent lower previsions

Het hoofdonderwerp van dit werk is het afleiden, voorstellen en bestuderen van voorspellende en parametrische gevolgtrekkingsmodellen die gebaseerd zijn op de theorie van coherente onderprevisies. Een belangrijk nevenonderwerp is het vinden en bespreken van extreme onderwaarschijnlijkheden. In het hoofdstuk ‘Modeling uncertainty’ geef ik een inleidend overzicht van de theorie van coherente onderprevisies ─ ook wel theorie van imprecieze waarschijnlijkheden genoemd ─ en de ideeën waarop ze gestoeld is. Deze theorie stelt ons in staat onzekerheid expressiever ─ en voorzichtiger ─ te beschrijven. Dit overzicht is origineel in de zin dat ze meer dan andere inleidingen vertrekt van de intuitieve theorie van coherente verzamelingen van begeerlijke gokken. Ik toon in het hoofdstuk ‘Extreme lower probabilities’ hoe we de meest extreme vormen van onzekerheid kunnen vinden die gemodelleerd kunnen worden met onderwaarschijnlijkheden. Elke andere onzekerheidstoestand beschrijfbaar met onderwaarschijnlijkheden kan geformuleerd worden in termen van deze extreme modellen. Het belang van de door mij bekomen en uitgebreid besproken resultaten in dit domein is voorlopig voornamelijk theoretisch. Het hoofdstuk ‘Inference models’ behandelt leren uit monsters komende uit een eindige, categorische verzameling. De belangrijkste basisveronderstelling die ik maak is dat het bemonsteringsproces omwisselbaar is, waarvoor ik een nieuwe definitie geef in termen van begeerlijke gokken. Mijn onderzoek naar de gevolgen van deze veronderstelling leidt ons naar enkele belangrijke representatiestellingen: onzekerheid over (on)eindige rijen monsters kan gemodelleerd worden in termen van categorie-aantallen (-frequenties). Ik bouw hier op voort om voor twee populaire gevolgtrekkingsmodellen voor categorische data ─ het voorspellende imprecies Dirichlet-multinomiaalmodel en het parametrische imprecies Dirichletmodel ─ een verhelderende afleiding te geven, louter vertrekkende van enkele grondbeginselen; deze modellen pas ik toe op speltheorie en het leren van Markov-ketens. In het laatste hoofdstuk, ‘Inference models for exponential families’, verbreed ik de blik tot niet-categorische exponentiële-familie-bemonsteringsmodellen; voorbeelden zijn normale bemonstering en Poisson-bemonstering. Eerst onderwerp ik de exponentiële families en de aanverwante toegevoegde parametrische en voorspellende previsies aan een grondig onderzoek. Deze aanverwante previsies worden gebruikt in de klassieke Bayesiaanse gevolgtrekkingsmodellen gebaseerd op toegevoegd updaten. Ze dienen als grondslag voor de nieuwe, door mij voorgestelde imprecieze-waarschijnlijkheidsgevolgtrekkingsmodellen. In vergelijking met de klassieke Bayesiaanse aanpak, laat de mijne toe om voorzichtiger te zijn bij de beschrijving van onze kennis over het bemonsteringsmodel; deze voorzichtigheid wordt weerspiegeld door het op deze modellen gebaseerd gedrag (getrokken besluiten, gemaakte voorspellingen, genomen beslissingen). Ik toon ten slotte hoe de voorgestelde gevolgtrekkingsmodellen gebruikt kunnen worden voor classificatie door de naïeve credale classificator.This thesis's main subject is deriving, proposing, and studying predictive and parametric inference models that are based on the theory of coherent lower previsions. One important side subject also appears: obtaining and discussing extreme lower probabilities. In the chapter ‘Modeling uncertainty’, I give an introductory overview of the theory of coherent lower previsions ─ also called the theory of imprecise probabilities ─ and its underlying ideas. This theory allows us to give a more expressive ─ and a more cautious ─ description of uncertainty. This overview is original in the sense that ─ more than other introductions ─ it is based on the intuitive theory of coherent sets of desirable gambles. I show in the chapter ‘Extreme lower probabilities’ how to obtain the most extreme forms of uncertainty that can be modeled using lower probabilities. Every other state of uncertainty describable by lower probabilities can be formulated in terms of these extreme ones. The importance of the results in this area obtained and extensively discussed by me is currently mostly theoretical. The chapter ‘Inference models’ treats learning from samples from a finite, categorical space. My most basic assumption about the sampling process is that it is exchangeable, for which I give a novel definition in terms of desirable gambles. My investigation of the consequences of this assumption leads us to some important representation theorems: uncertainty about (in)finite sample sequences can be modeled entirely in terms of category counts (frequencies). I build on this to give an elucidating derivation from first principles for two popular inference models for categorical data ─ the predictive imprecise Dirichlet-multinomial model and the parametric imprecise Dirichlet model; I apply these models to game theory and learning Markov chains. In the last chapter, ‘Inference models for exponential families’, I enlarge the scope to exponential family sampling models; examples are normal sampling and Poisson sampling. I first thoroughly investigate exponential families and the related conjugate parametric and predictive previsions used in classical Bayesian inference models based on conjugate updating. These previsions serve as a basis for the new imprecise-probabilistic inference models I propose. Compared to the classical Bayesian approach, mine allows to be much more cautious when trying to express what we know about the sampling model; this caution is reflected in behavior (conclusions drawn, predictions made, decisions made) based on these models. Lastly, I show how the proposed inference models can be used for classification with the naive credal classifier

Un enfoque de sustentabilidad utilizando lógica difusa y minería de datos

[ES] Sustainable development goals are now the agreed criteria to monitor states, and this work will demonstrate that numerical and graphical methods are valuable tools in assessing progress. Fuzzy Logic is a reliable procedure for transforming human qualitative knowledge into quantitative variables that can be used in the reasoning of the type “if, then” to obtain answers pertaining to sustainability assessment. Applications of machine learning techniques and artificial intelligence procedures span almost all fields of science. Here, for the first-time, unsupervised machine learning is applied to sustainability assessment, combining numerical approaches with graphical procedures to analyze global sustainability. CD HJ-Biplots to portray graphically the sustainability position of a large number of countries are a useful complement to mathematical models of sustainability. Graphical information could be useful to planners it shows directly how countries are grouped according to the most related sustainability indicators. Thus, planners can prioritize social, environmental, and economic policies and make the most effective decisions. One could graphically observe the dynamic evolution of sustainability worldwide over time with a graphical approach used to draw relevant conclusions. In an era of climate change, species extinction, poverty, and environmental migration, such observations could aid political decision-making regarding the future of our planet. A large number of countries remain in the areas of moderate or low sustainability. Fuzzy logic has proven to be an uncontested numerical method as it occurs with SAFE. An unsupervised learning method called Variational Autoencoder interplay Graphical Analysis (VEA&GA) has been proposed, to support sustainability performance with appropriate training data. The promising results show that this can be a sound alternative to assess sustainability, extrapolating its applications to other kinds of problems at different levels of analysis (continents, regions, cities, etc.) further corroborating the effectiveness of the unsupervised training methods